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M Sajid, T Hayat (2008)
Comparison of HAM and HPM methods in nonlinear heat conduction and convection equationsNonlin. Anal. B: Real World Appl., 9
A Mastroberardino (2011)
Homotopy analysis method applied to electrohydrodynamic flowCommun. Nonlin. Sci. Numer. Simul., 16
C Liu (2010)
The essence of the generalized Newton binomial theoremCommun. Nonlin. Sci. Numer. Simul., 15
I Hashim, O Abdulaziz, S Momani (2009)
Homotopy analysis method for fractional IVPsCommun. Nonlin. Sci. Numer. Simul., 14
S Liao (2010)
An optimal homotopy-analysis approach for strongly nonlinear differential equationsCommun. Nonlin. Sci. Numer. Simul., 15
SJ Liao (2012)
Homotopy Analysis Method in Nonlinear Differential Equations
S Liang, S Liu (2014)
An open problem on the optimality of an asymptotic solution to Duffing’s nonlinear oscillation problemCommun. Nonlin. Sci. Numer. Simul., 19
(2003)
Homotopy perturbation method: a new nonlinear analytical techniqueAppl. Math. Comput., 135
SJ Liao (2003)
Beyond Perturbation and Introduction to the Homotopy Analysis Method
SJ Liao (2004)
On the homotopy analysis method for nonlinear problemsAppl. Math. Comput., 147
DD Ganji (2006)
The application of He’s homotopy perturbation method to nonlinear equations arising in heat transferPhys. Lett. A, 355
S Abbasbandy (2006)
Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition methodAppl. Math. Comput., 172
R Ellahi, M Raza, K Vafai (2012)
Series solutions of non-Newtonian nanofluids with Reynolds’ model and Vogel’s model by means of the homotopy analysis methodMath. Comput. Model., 55
C Liu (2010)
The essence of the homotopy analysis methodAppl. Math. Comput., 216
C Liu (2011)
The essence of the generalized Taylor theorem as the foundation of the homotopy analysis methodCommun. Nonlin. Sci. Numer. Simul., 16
Y Tan, S Abbasbandy (2008)
Homotopy analysis method for quadratic Riccati differential equationCommun. Nonlin. Sci. Numer. Simul., 13
In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method (HAM). In the present paper, we demonstrate that such a view is only valid in very special cases, and in general, the HAM is far more robust. In particular, the equivalence is only valid when the solution is represented as a power series in the independent variable. As has been shown many times, alternative basis functions can greatly improve the error properties of homotopy solutions, and when the base functions are not polynomials or power functions, we no longer have that the generalized Taylor series approach is equivalent to the HAM. In particular, the HAM can be used to obtain solutions which are global (defined on the whole domain) rather than local (defined on some restriction of the domain). The HAM can also be used to obtain non-analytic solutions, which by their nature can not be expressed through the generalized Taylor series approach. We demonstrate these properties of the HAM by consideration of an example where the generalizes Taylor series must always have a finite radius of convergence (and hence limited applicability), while the homotopy solution is valid over the entire infinite domain. We then give a second example for which the exact solution is not analytic, and hence, it will not agree with the generalized Taylor series over the domain. Doing so, we show that the generalized Taylor series approach is not as robust as the HAM, and hence, the HAM is more general. Such results have important implications for how iterative solutions are calculated when approximating solutions to nonlinear differential equations.
Numerical Algorithms – Springer Journals
Published: Dec 26, 2016
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